Title: Quantitative estimates on singularities
of minimal hypersurfaces
Abstract: We will discuss the
occasionally unavoidable presence of singularities on
minimal hypersurfaces in high dimensional ambient spaces and
estimates on its size. This is seemingly an analysis problem
but the variational notion of minimal hypersurfaces plays a
much more important role than its defining PDE. The proof
relies simply on coverings arguments by suitable open sets
and we will go over the main ideas and consequences.
Date: Tuesday, 25th March 2025,
09:00-10:00 TST (GMT+8) , online on Zoom
Title: Minimal hypersurfaces: bubble convergence and index
Abstract:
The regularity theories of Schoen--Simon--Yau and Schoen--Simon for stable minimal hypersurfaces are foundational in geometric analysis. Using this regularity theory, in low dimensions, Chodosh--Ketover--Maximo, and Buzano--Sharp, studied singularity formation along sequences of minimal hypersurfaces through a bubble analysis.
I will review this background, before talking about my recent work in this bubble analysis theory. In particular I will show how to obtain upper semicontinuity of index plus nullity along a bubble converging sequence of minimal hypersurfaces.
Date: Tuesday, 1st April 2025,
09:00-10:00 TST (GMT+8) , online on Zoom
Speaker:Dr.
Zane Li (North Carolina State University)
Title: Mixed norm decoupling for paraboloids
Abstract:
In this talk we discuss mixed
norm decoupling estimates for the paraboloid. One motivation
of considering such an estimate is a conjectured mixed norm
Strichartz estimate on the torus which essentially is an
estimate about exponential sums. This is joint work with
Shival Dasu, Hongki Jung, and José Madrid.
Tuesday, 8th April 2025,
09:00-10:00 TST (GMT+8) , in Room M210 in NTNU Gongguan Campus
Mathematics Building, also online on ZoomSpeaker:Dr. Alan Chang (Washington
University in St. Louis)
Title: Venetian blinds, digital sundials, and
efficient coverings
Abstract:
Davies's efficient covering
theorem states that we can cover any measurable set in the plane by
lines without increasing the total measure. This result has a dual
formulation, known as Falconer's digital sundial theorem, which
states that we can construct a set in the plane to have any desired
projections, up to null sets. The argument relies on a Venetian
blind construction, a classical method in geometric measure theory.
In joint work with Alex McDonald and Krystal Taylor, we study a
variant of Davies's efficient covering theorem in which we replace
lines with curves. This has a dual formulation in terms of nonlinear
projections.
Date:
Tuesday, 15th April 2025,
09:00-10:00 TST (GMT+8) , online on Zoom Speaker:
Dr. Bochen Liu (Southern University of Science and
Technology) Title: TBA
Abstract: TBA